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| Mirrors > Home > MPE Home > Th. List > ttrclco | Structured version Visualization version GIF version | ||
| Description: Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.) |
| Ref | Expression |
|---|---|
| ttrclco | ⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relres 6003 | . . . 4 ⊢ Rel (𝑅 ↾ V) | |
| 2 | ssttrcl 9737 | . . . 4 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)) | |
| 3 | coss2 5847 | . . . 4 ⊢ ((𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) → (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V))) | |
| 4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) |
| 5 | ttrcltr 9738 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) | |
| 6 | 4, 5 | sstri 3973 | . 2 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) |
| 7 | relco 6106 | . . . 4 ⊢ Rel (t++(𝑅 ↾ V) ∘ 𝑅) | |
| 8 | dfrel3 6198 | . . . 4 ⊢ (Rel (t++(𝑅 ↾ V) ∘ 𝑅) ↔ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ 𝑅)) | |
| 9 | 7, 8 | mpbi 230 | . . 3 ⊢ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ 𝑅) |
| 10 | resco 6250 | . . 3 ⊢ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) | |
| 11 | ttrclresv 9739 | . . . 4 ⊢ t++(𝑅 ↾ V) = t++𝑅 | |
| 12 | 11 | coeq1i 5850 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ 𝑅) = (t++𝑅 ∘ 𝑅) |
| 13 | 9, 10, 12 | 3eqtr3i 2765 | . 2 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) = (t++𝑅 ∘ 𝑅) |
| 14 | 6, 13, 11 | 3sstr3i 4014 | 1 ⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1539 Vcvv 3463 ⊆ wss 3931 ↾ cres 5667 ∘ ccom 5669 Rel wrel 5670 t++cttrcl 9729 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pr 5412 ax-un 7737 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-ttrcl 9730 |
| This theorem is referenced by: (None) |
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